# Ortoedres amb longitud d'arestes enteres / Cuboids with integer length edges

## TL;DR

This paper develops an iterative method to count the number of distinct cuboids with integer edges and volume N, providing explicit formulas for certain cases and extending the analysis to rectangles made of squares.

## Contribution

It introduces a new iterative approach to compute the number of cuboids with integer edges for any volume N, including explicit formulas for specific factorizations.

## Key findings

- Derived an iterative method for calculating (N)
- Provided explicit formulas for (N) when N is a product of two prime powers
- Extended analysis to count rectangles formed by squares with a given area

## Abstract

In this article we study the number of different cuboids $\mathcal{O}(N)$ that can be built with an arbitrary number $N$ of equal cubes. This problem is equivalent to find the number of different cuboids of volume $N$ with integer length edges. We obtain an iterative method to calculate the value of $\mathcal{O}(N)$ for any $N$. Using this method we obtain an explicit formula when $N$ is the product of two powers of prime numbers. The bidimensional case is also studied and we give a general formula to determine the number of different rectangles that can be built with an arbitrary number of equal squares.